Question

Let R be a relation from N to N defined by R = [(a, b) : a, b ∈ N and a = b²].
Are the following statements true?

(i) (a, a) ∈ R for all a ∈ N
(ii) (a, b) ∈ R ⇒ (b, a) ∈ R
(iii) (a, b) ∈ R and (b, c) ∈ R ⇒ (a, c) ∈ R

Answer 1

Given: R = [(a, b) : a, b ∈ N and a = b²]

(i) (a, a) ∈ R for all a ∈ N.
Here, 2 ∈N, but $2\ne 2^2$.
∴ (2,2)$\notin$R
False

(ii) (a, b) ∈ R ⇒ (b, a) ∈ R
∵ 4 = 2²
(4, 2) ∈ R, but (2,4)$\notin$R.
False

(iii) (a, b) ∈ R and (b, c) ∈ R ⇒ (a, c) ∈ R
∵ 16 = 4² and 4 = 2²
∴ (16, 4) ∈ R and (4, 2) ∈ R
Here,
(16,2)$\notin$R
False